3D Rotation
3D Rotation#
Rz#

$ begin{bmatrix} x & y & z & 1 end{bmatrix} begin{bmatrix} costheta & sintheta & 0 & 0 \ -sintheta & costheta & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix} $
Demo#

$ p = (1,0,0) \ theta=90 \ {p}’ = (0,1,0) $
$ begin{bmatrix} 1 & 0 & 0 & 1 end{bmatrix} begin{bmatrix} 0 & 1 & 0 & 0 \ -1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix}= begin{bmatrix} 0 & 1 & 0 & 1 end{bmatrix} $
Rx#
$ begin{bmatrix} x & y & z & 1 end{bmatrix} begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & costheta & sintheta & 0 \ 0 & -sintheta & costheta & 0 \ 0 & 0 & 0 & 1 end{bmatrix} $
Demo#

$ p = (0, 1, 0) \ theta = 90 \ {p}’ = (1, 0, 0) $
$ begin{bmatrix} 0 & 1 & 0 & 1 end{bmatrix} begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & 0 & 1 end{bmatrix}= begin{bmatrix} 0 & 0 & 1 & 1 end{bmatrix} $
Ry#
$ begin{bmatrix} x & y & z & 1 end{bmatrix} begin{bmatrix} costheta & 0 & -sintheta & 0 \ 0 & 1 & 0 & 0 \ sintheta & 0 & costheta & 0 \ 0 & 0 & 0 & 1 end{bmatrix} $
Demo#

$ p = (0, 0, 1) \ theta = 90 {p}’ = (1, 0, 0) $
$ begin{bmatrix} 0 & 0 & 1 & 1 end{bmatrix} begin{bmatrix} 0 & 0 & -1 & 0 \ 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 end{bmatrix}= begin{bmatrix} 1 & 0 & 0 & 1 end{bmatrix} $